The Indefinite Integral and Basic Rules of Integration

Antiderivatives and the Indefinite Integral

Let a function f (x) be defined on some interval I. The function F (x) is called an antiderivative of f (x), if

\[F^\prime\left( x \right) = f\left( x \right)\]

for all x in the interval I.

There is an infinite number of antiderivatives of a function f (x), all differing only by a constant C:

\[\left( {F\left( x \right) + C} \right)^\prime = F^\prime\left( x \right) + C^\prime = f\left( x \right) + 0 = f\left( x \right).\]

The set of all antiderivatives for a function f (x) is called the indefinite integral of f (x) and is denoted as

\[{\int} {{f\left( x \right)}{dx}} = F\left( x \right) + C,\;\;\text{if}\;\;F^\prime\left( x \right) = f\left( x \right).\]

In this definition, the ∫ is called the integral symbol, f (x) is called the integrand, x is called the variable of integration, dx is called the differential of the variable x, and C is called the constant of integration.

Indefinite Integral of Some Common Functions

Integration is the reverse process of differentiation, so the table of basic integrals follows from the table of derivatives.

It is supposed here that \(a,\) \(p\left( {p \ne 1} \right),\) \(C\) are real constants, \(b\) is the base of the exponential function \(\left( {b \ne 1, b \gt 0} \right).\)

Properties of the Indefinite Integral

If \(a\) is some constant, then

\[\int {af\left( x \right)dx} = a\int {f\left( x \right)dx},\]

i.e. the constant coefficient can be carried outside the integral sign.

For functions \(f\left( x \right)\) and \(g\left( x \right),\)

\[\int {\left[ {f\left( x \right) \pm g\left( x \right)} \right]dx} = \int {f\left( x \right)dx} \pm \int {g\left( x \right)dx} ,\]

i.e. the indefinite integral of the sum (difference) equals to the sum (difference) of the integrals.

Calculation of integrals using the linear properties of indefinite integrals and the table of basic integrals is called direct integration.

Solved Problems

Example 1.

Evaluate the indefinite integral \[\int {\left( {3{x^2} – 6x + 2\cos x} \right)dx} .\]